Magnetic metasurfaces properties in the near field regions

In this paper, we present a general equivalent-circuit interpretation of finite magnetic metasurfaces interacting with an arbitrary arrangement of RF coils operating in near-field regime. The developed model allows to derive a physical interpretation of the interactions between the metasurface and the surrounding RF coils, both transmitting and receiving. Indeed, especially for near-field applications, the metasurface presence modifies the behavior of each RF coil differently, due to the specific reciprocal interactions. Hence, the proposed approach introduces a source-related complex magnetic permeability matrix, overcoming the traditional bulk definition. To prove the model validity against full-wave simulations, we present two significant test cases, commonly used in practical applications. The former is represented by the simple metasurface-coil arrangement from which important and fundamental considerations can be drawn. The latter system is composed by a transmitting and a receiving coil with a metasurface in between; detailed explanations on the metasurface interactions with both the RF coils are developed. Finally, we also achieved an excellent agreement between the numerical results and the measurements obtained through fabricated prototypes. In summary, the circuit interpretation herein presented, in addition to the rigorous electromagnetic theoretical approaches already appeared in the open literature, reveals useful in providing quantitative, practical, and easy-to-handle guidelines for the design and physical understanding of finite magnetic metasurfaces interacting with arbitrary RF coils arrangements in the near-field regime.

of resonant unit-cells, like spiral or split-ring resonators (Fig. 1a). The whole array reacts to an impinging magnetic field with a resonant behavior (usually described through a Lorentzian model), making possible to exploit enhanced and μ-negative permeability at specific bandwidths. It has been demonstrated in 37 that the entire metasurface can be represented by an equivalent RLC model; this step is fundamental to describe and quantify its interactions with other RF coils, as it will be shown in the following sections. A brief recall of the metasurface RLC reduction is herein reported for the reader clarity: more details can be found in 37 . In a generic arrangement, we can have M fed RF coils interacting with a passive metasurface. The metasurface can be assumed to be formed by N × N = P resonant unit-cells. If we refer to the RF coils with the first M indices and with the following P indices to the elements of the array, the overall system impedance matrix can be written as below. where c i is the generic i-th complex current coefficient and I x is the equivalent current flowing in the RLC model of the array. By summing up equations from row M + 1 to row M + P and re-arranging terms, it is possible to write the following system, where the P elements of the metasurface have been substituted by their equivalent resonator (marked with index x).
In particular, the Z xx term can be interpreted as the self-impedance of the metasurface equivalent resonator (RLC series), whereas I x is its equivalent flowing current and the various Z xi terms correspond to the mutual coupling coefficients between the metasurface and each of the M RF coils 37 .
At this point, we can express the current I x that flows in the equivalent metasurface RLC circuit as a function of the other M RF coils currents, exploiting the equations system (3): Thus, we can substitute the expression (4) in the first M equations of (3); therefore, the effect of the metasurface presence over the other M RF coils can be easier highlighted: Further, we can write the generic element of the impedance matrix in (5) by introducing the source-related complex (relative) magnetic permeability value µ ij r , as described below.
As it will be better clarified for the adopted test-cases, each RF coil undergoes to a unique impedance modification due to the presence of the magnetic metasurface, in dependence of its relative position and interactions with other elements. Thus, each RF coil, and each corresponding mutual coupling term of the impedance matrix (5), experiences a different equivalent complex permeability value (what we call source-related permeability, Fig. 1b). Finally, the overall M RF coils can be represented by the following complete equations system, where the metasurface presence has been translated into the complex (relative) magnetic permeability coefficients µ ij r : Therefore, practical guidelines and physical interpretations useful to accomplish the desired design can be derived from the retrieved lumped elements of the entire system, by using the complex relative permeability matrix µ r . Indeed, additional degrees of freedom are available to the designer to optimize the M RF coils system, consequently exploiting more effectively its potentialities through the introduced source-related magnetic permeability values.

Selected experimental set-ups.
It is worth remarking that the aim of this paper is to develop a circuit-based model able to provide useful and practical design guidelines for the realization of a finite magnetic metasurface interacting with a generic RF coils arrangement, also giving a physical interpretation of the entire system by using the retrieved complex magnetic permeability matrix, as previously explained. Therefore, we herein report two meaningful test-cases adopted to validate the proposed approach. Firstly, a single RF coilmetasurface system is faced; this simple configuration can be seen as the building block of several applications, as for instance in Magnetic Resonance Imaging RF coil design 38 . Secondly, the system formed by a transmitting coil, a metasurface and a receiving coil is analyzed with our circuit model (as schematically depicted in Fig. 1a): for this case, some important and effective design considerations can be drawn, especially suitable for resonant inductive Wireless Power Transfer applications 19 . Nonetheless, the provided analysis is completely general and can be applied also to more complex coils arrangements. Specifically, we exploited a Method of Moments electromagnetic solver (Feko suite, Altair, Troy, MI, USA) for the entire design process while the measurements have been performed by using the Keysight (Santa Rosa, CA, USA) N9918B FieldFox Handheld Vector Network Analyzer.
Single coil-metasurface system description. The first proposed test-case is depicted in Fig. 2a. It comprises an RF active planar spiral with a 10 cm external diameter. The coil presents 5 turns of a 28 AWG lossy copper wire, with a pitch between adjacent branches of 0.68 mm. No additional reactive loads are added, and the spiral is non-resonant.
We also consider a metasurface made of a planar 3 × 3 structure; each unit-cell is an 8-turns passive resonant spiral with a 2.4 cm external diameter. The unit-cell pitch is 0.18 mm, made of 28 AWG lossy copper wire. The overall metasurface is positioned 5 mm away from the active RF coil, in a coaxial fashion. In order to operate at the desired working frequency (around 6 MHz), a 390 pF capacitor is added in series to each unit-cell. The choice of the working frequency is arbitrary and other values might have been chosen as well. Following the methodology reported in 37 , we extracted the equivalent RLC model of the metasurface together with the mutual coupling coefficient with the active RF coil. The obtained values are: R meta = 4.13 Ω, L meta = 14.97 μH, C meta = 43.35 pF, M meta-coil = 2.09 μH.
Besides the numerical simulations, we also fabricated prototypes to perform experimental measurements (Fig. 3a, b). The prototypes are built with a 28 AWG copper wire glued onto an 0.8 mm thick FR4 slab (ε r = 4.3, tanδ = 0.02). The capacitors are soldered on the other side, following the design specifications. In addition, Fig. 3c shows the final experimental arrangement, where a plastic framework is employed to precisely positioning the radiating elements in terms of distances, exploiting the 4 external holes drilled on the FR4 substrate.
Transmitter-metasurface-receiver system description. The second CAD configuration is shown in Fig. 2b. Essentially, it consists in the same configuration of the previous test-case but an additional RF passive coil is added. This coil is geometrically identical to the fed RF spiral and it is non-resonant (thus, it is not loaded with capacitors). It has been placed 10 cm away from the fed one, always in a coaxial fashion. This arrangement is typically used in inductive WPT, where a transmitting coil, a metasurface and a receiving coil are positioned as in this example.
As in the previous case, we also extracted the mutual coupling coefficient between the metasurface and the added receiving coil. The other lumped values, i.e. the metasurface equivalent RLC and its mutual coupling with the fed coil, have been already calculated from the first configuration. The coefficient M meta-receiver was estimated equal to 0.12 μH.
Finally, also for this test-case, the experimental set-up was arranged (Fig. 3d).  where the RF coil is indicated with the index 1, whereas the metasurface is globally reduced to its single equivalent resonator and pointed out with index 2. By expressing the current I 2 as a function of I 1 , it is straightforward writing down the impedance seen at the port 1: We can now exploit the developed analytical model to elaborate equation (9); in particular, we assume that the RF coil (element 1) is not loaded with any capacitor, thus it is represented by its self-resistance and inductance: Through some algebraic manipulations, we can express the port impedance in the following form: At this point, we can introduce the source-related complex (relative) magnetic permeability μ r ; this permeability is associated to the equivalent medium in which the RF coil 1 is immersed (Fig. 1b): and, thus, we can express this equivalent complex relative permeability as a function of the lumped elements of our circuit equivalent model: www.nature.com/scientificreports/ In order to report the complex magnetic permeability behavior versus frequency expressed by the Eq. (13), we used the lumped elements values retrieved from the CAD model described in Fig. 2a; the results are shown in Fig. 4a ( µ ′ r ) and Fig. 4(b) ( µ ′′ r ). In these graphs, we also compared the pure analytically retrieved permeability against full-wave simulations and experimental measurements. As evident from Fig. 4, we observe an excellent agreement, thus demonstrating the reliability of the circuit model. It may be worth highlighting that this is the equivalent magnetic permeability as seen by the RF coil 1 itself; thus, it does not represent the actual bulk permeability of the metasurface alone. Hence, differently from the canonical approach, we avoid describing the bulk permeability of the proposed metasurface; instead, the metasurface equivalent effect onto the medium surrounding the RF coils arrangement is pointed out (from which the term source-related permeability).
In this sense, a noticeable result that has been proved in the literature is that a μ r = −1 metamaterial can enhance the evanescent magnetic field produced by an RF coil 15,30,38 . Hence, the question is how a metamaterial with a μ r = −1 as its own bulk permeability interacts with the RF coil from a circuital point of view. As typically presented in the literature, a metamaterial can be simulated by a numerical solver and it consists of a thick slab of homogeneous material showing the desired permeability. As a matter of fact, the slab thickness is often  www.nature.com/scientificreports/ larger than the diameter of the RF coil placed in its proximity 30,38 (see Fig. 5). In addition, it is also positioned very close to the coil. Since the electromagnetic field produced by a resonator significantly drops for distances larger than its diameter 7 , this configuration corresponds to divide the space where the RF coil is placed into two subdomains: a homogenous material with μ r = −1 at one side and free space on the other side (μ r = +1) (Fig. 5). Thus, it is reasonable to expect that the effective magnetic permeability seen by the RF coil will be the average value of the permeability of the two subdomains; this implies that the equivalent medium permeability is zero in its real component (see (13)). Therefore, according to (12), this condition has the effect of cancelling the reactive component of the RF coil impedance, thus putting the coil under resonance. By referring to the results of Fig. 4, the zero value for permeability happens at f = 6.4 MHz. Hence, the current flowing in the RF coil dramatically increases for a given voltage excitation; this is consistent with what observed in the literature and predicted by the theoretical derivations based on Maxwell equations 30 . In particular, Fig. 6 reports the H-field maps obtained for the CAD model of Fig. 2a through full-wave simulations, without and in the presence of the metasurface at the μ r = −1 point. By forcing the same circulating current in the RF coil for both the configurations, it is evident how the metasurface is able to enhance the H-field produced by the driving coil. Therefore, the herein provided circuital model is able to describe the μ r = −1 condition only through the retrieved lumped parameters (13); thus, the synthesis of artificial materials can be greatly simplified.
In a practical set-up, a metasurface is fabricated by starting from a 2D array of resonant magnetic inclusions, like spiral or split ring resonators. As a matter of fact, the actual thickness of the realized metasurface is not directly correlated to the equivalent thickness of the homogeneous material adopted in full-wave simulations (Fig. 5), being extremely thinner (usually a few millimeters). As evident in (13), the retrieved permeability can be finely tailored on the basis of the lumped elements values described in our model. Hence, the metasurface has an effective thickness that can be modulated through the lumped model. In fact, the availability of a simple and straightforward circuit model, in which the relative lumped elements can be modified to shape the metasurface magnetic response according to the design requirements, is one of the major advantages of the proposed approach. In particular, we can easily adjust such parameters by noticing that M 12 , L 2 , C 2 and R 2 are quantities ruled by the proposed model. Therefore, by modifying the distance between metasurface and RF coil (M 12 ), the unit-cell design (to control R 2 , L 2 and C 2 ) and their relative position (i.e., the array periodicity), we can obtain the desired curve for the equivalent permeability experienced by the RF coil. This implies that the proposed circuit model can also be used to characterize intermediate situations, in which, for instance, the metamaterial cannot be approximated by the semi-infinite hypothesis represented in Fig. 5. In that case, the equivalent permeability seen by the RF coil will be the average value between the air (present on one side) and an equivalent material with a diluted permeability. Several models in the literature have been developed to describe similar situations, but typically considering only the dielectric counterpart 7,39 . In this regard, Fig. 7 reports some meaningful examples of real and imaginary permeability values retrieved with the analytical model for the proposed radiating configuration. In particular, in Fig. 7a, b, the distance between the RF coil and the metasurface is varied, from 5 mm to 11 mm; this implies that the mutual coupling M 12 between the RF coil and the metasurface is becoming smaller with increasing distances and, as predicted, the complex permeability amplitude related to the RF coil accordingly decreases, simulating a progressively thinner metamaterial. Additionally, in Fig. 7c, d, the analytical model is employed to retrieve the source related complex permeability when the metasurface unit-cell capacitive load is gradually changed, from 351 pF to 429 pF; as evident, the complex permeability experienced by the RF coil can be modulated and controlled, on the basis of the specific application requirements.
For instance, in 40 it is reported that a metasurface with a pure imaginary permeability is able to perform as an ideal microwave absorber; whereas all the mathematical analysis is therein performed under the plane wave hypothesis, we can completely overcome this limit and design arbitrary metasurface complex permeabilities also (a) (b) Figure 6. Numerical magnetic field maps evaluated for the configuration shown in the inset on a plane perpendicular to the metasurface (xz plane in Fig. 2a): actively fed RF coil without (a) and with (b) metasurface, in the condition of the same circulating current. As evident, the metasurface presence with a μ r = − 1 behavior is able to significantly enhance the magnetic field amplitude, in according to the theoretical model. www.nature.com/scientificreports/ considering near-field sources. This latter condition is generally closer to practical applications, especially those at relatively low operative frequency.
As an added value, the same metasurface can be even used to compensate any desired reactance of the coil. Before its resonant frequency, when the real permeability is positive, capacitive reactance can be compensated; conversely, after the resonant point, a negative value of the real permeability can be used to null an inductive impedance (Fig. 4a, b). Moreover, provided that the permeability imaginary component, introduced by the metasurface ohmic losses, retains the proper value to guarantee a good matching to the port impedance (12), not only the tuning of the RF coil (i.e., the cancellation of its reactive impedance component), but also the matching to the output impedance of a generator can be achieved (for instance, 50 Ω). To this aim, Fig. 8 reports both the numerical and the experimental S 11 parameter of the model described in Fig. 2a Indeed, by a proper metasurface design, we can achieve tuning and matching of an RF coil without using any capacitive load and/or matching network. This implies a more efficient design of the RF coil, avoiding the use of lumped capacitors that are often the cause of undesirable electric field hot spots 20 .  www.nature.com/scientificreports/ Transmitter-metasurface-receiver system. By recurring to the same circuital model previously described, it is possible to express the equations system for the CAD in Fig. 2b in the following way: in which we denote with the indices 1 and 2 the fed transmitter and the passive receiver coil, respectively; in this case, the magnetic metasurface has been replaced by its equivalent resonator and addressed with index 3. We can now proceed in the same fashion as for the previous case, i.e. we express the metasurface equivalent current I 3 as a function of I 1 and I 2 and substitute it in the first two equations of (14). The result is a 2-port system whose impedance matrix has the following form: From (15), it is evident that both the transmitter and receiver self-impedances are influenced by the metasurface presence. Indeed, all the 4 terms of (15) contain a dependence on the metasurface self-impedance Z 33 at the denominator, thus presenting a peak at its resonance. It is easy to verify that an expression formally equivalent to Eq. (13) can be derived for both the transmitter and receiver. Thus, by exploiting the single unit-cell design (R 3 , L 3 , C 3 ), the cell periodicity within the array and the metasurface distance with the RF coils (M 13 /M 23 terms), it is possible to manipulate both the reactive and the real components of the RF coils self-impedances. Following the model developed for the single coil-metasurface case, it is worth pointing out that both transmitter and receiver experience different magnetic permeabilities; hence, it immediately emerges that a magnetic metasurface acts differently on the RF coils constituting the system, depending on its relative position and on the coils geometrical constraints, as theoretically predicted. In particular, the transmitter-related complex permeability is coincident with the behavior reported in Fig. 4a, b; conversely, the receiver permeability is shown in Fig. 9a, b. It is apparent from the permeability values that the receiver is minimally affected by the metasurface presence; this is coherent with the greater distance that separates the receiver from the metasurface with respect to the transmitter (i.e., 95 mm against 5 mm).
To report a practical scenario, in resonant inductive Wireless Power Transfer (WPT), the inductive coupling is exploited to transfer energy from an active RF coil towards a passive receiving RF coil; consequently, the most important term to be studied is the off-diagonal one in (15). Indeed, the effective Z 11 eff and Z 22 eff (the global self-impedances of transmitter and receiver) can always be compensated by resorting to a matching network or exploiting the transmitter and receiver distances with the metasurface as an additional design parameter 15,41 . Therefore, it is worth expressing the mutual coupling term Z 12 eff in its complete form to understand some interesting features on how a magnetic metasurface interacts and modifies an inductive link. Hence, we can write: www.nature.com/scientificreports/ where jωM 12 is the classical inductive mutual coupling term between the two RF coils (Z 12 ), in this case the transmitter and the receiver. Instead, the other additional term arises because of the metasurface presence, which is described through its equivalent resonator. By manipulating the above expression, we can directly express the source related magnetic permeability of the inductive link as: where we have assumed that the total mutual coupling between transmitter and receiver can be expressed as: In Fig. 9c, d we reported the real and imaginary component of this permeability, comparing the pure analytical solution against full wave simulations and experimental measurements, obtained from the set-up depicted in Fig. 2b. Again, we observe an excellent agreement among analytical model, full-wave simulations and measurements, thus demonstrating the accuracy of the equivalent circuit in effectively representing the real scenario.
At this point, from the graphs of Fig. 9c, d, some important observations can be drawn. We immediately reveal that the metasurface is able to eliminate, almost perfectly, the mutual coupling jωM 12 between transmitter and receiver. This happens slightly beyond the metasurface resonant point at f = 6.6 MHz (zero point cross of the real part of the retrieved permeability).
If the loss component of the retrieved permeability is low, then the metasurface acts as a perfect magnetic shield between the RF coils; indeed, the off-diagonal terms in (15) are nulled and the transmitter and receiver are decoupled. This effect has been observed in the literature and already exploited in different technological areas, as an alternative solution to ferrite shields for low frequency magnetic fields or in MRI array elements decoupling 20,23 . Obviously, this operative point must be avoided if the application under study is the wireless energy transfer between the two RF coils.
On the other hand, in WPT applications, the best working frequency results to be at the metasurface selfresonance (f = 6.25 MHz in Fig. 9c, d), when the reactive component of Z 33 is nulled 19 and the off-diagonal term Z 12 eff is maximized. Indeed, in this configuration, the magnetic metasurface is acting as the intermediate coil of a classical 3-coil system 42 . Provided that the impedances at the port 1 and 2 (transmitter and receiver) can be appropriately compensated and matched, this operative point can lead to the maximum coupling between the two RF coils. Since efficiency is directly dependent on the square of the absolute Z 12 eff value 19 , this means reaching the maximum energy delivery.
These two practically interesting working conditions, i.e. shielding and power transfer configurations, have been also evaluated through full-wave simulations. In particular, Fig. 10a reports the magnetic field distribution Figure 10. Numerical magnetic field maps evaluated for the configuration shown in the inset on a plane perpendicular to the metasurface (xz plane in Fig. 2b), in the space between transmitting and receiving coils.
(a) Magnetic field distribution evaluated with the metasurface used as a magnetic field shield, at 6.6 MHz. (b) Same field distribution with the metasurface tuned to enhance the mutual coupling between transmitter and receiver, at 6.25 MHz. It must be noted that the comparison is performed with the same circulating current in the transmitter.  Fig. 2b for the geometrical reference system) between transmitting and receiving coils when the metasurface is employed as a magnetic field shield (at 6.6 MHz). Conversely, Fig. 10b describes the same geometrical configuration but with the metasurface tuned to enhance the mutual coupling between transmitter and receiver (at 6.25 MHz). It must be noticed that both these numerical experiments have been carried out for the same circulating current in the transmitting coil, to obtain a fair comparison. The obtained numerical results confirmed what theoretically expected in terms of field distribution. Certainly, the doubt that fabricating a magnetic metasurface can be more problematic with respect to a simple additional repeater coil can be raised. However, some peculiar characteristics of magnetic metasurfaces cannot be achieved by a single additional coil, like enhanced misalignment robustness 37 and electric field shielding 43 .
In conclusion, when a magnetic metasurface interacts with RF coils, understanding that each coil experiences a peculiar equivalent permeability, depending on its position and design geometries, is crucial. In this way, the various RF coils behaviors can be more easily manipulated, rather than retrieving the bulk magnetic properties of the metasurface itself, which is not convenient to describe near-field interactions. By expressing such interactions with an equivalent circuit, a straightforward and more effective design process can be accomplished, significantly aiding the engineering step, as summarized in the flow-chart scheme reported in Fig. 11.

Discussion
In this paper, we presented a general equivalent-circuit interpretation of finite magnetic metasurfaces interacting with an arbitrary arrangement of RF coils operating in near-field regime. In particular, the developed model is able to provide a useful physical understanding, for which the metasurface complex magnetic permeability can be appropriately engineered in dependence of the various RF coils constituting the overall system. It is worth mentioning that arbitrary RF coils arrangements interacting with the metasurface can be described and analyzed, hence making the model general and easily extendible for several different applications.
We first recalled how to reduce similar structures interacting with RF coils to their own equivalent resonator model, further analyzing how a magnetic metasurface affects differently the surrounding RF coils, defining a proper source-related complex relative magnetic permeability matrix. Afterwards, we deeply studied two meaningful test-cases to validate the proposed circuital model. Firstly, we faced the single coil-metasurface system, which is the simplest possible configuration but extremely interesting for the related practical implications; secondly, we studied the classical transmitter-metasurface-receiver set-up, typical of Wireless Power Transfer applications. We compared the analytical predictions with full-wave simulations, obtaining excellent results and, thus, demonstrating the reliability and accuracy of the circuit interpretation. Moreover, measurements performed over the fabricated prototypes reinforced the numerical conclusions.
Although very detailed theoretical works describing such structures through full Maxwell equations are already available in the literature, a lumped elements model can be extremely useful in practical design and engineering process. Indeed, the possibility to quantify and manipulate the key parameters of a system results in a major advantage from a design point of view in a large number of applications, like Wireless Power Transfer and Magnetic Resonance Imaging.
The circuit model herein presented is general and we foresee an extension to electric near-field interactions between generic antennas and metasurfaces configurations.   Figure 11. Design flowchart using the proposed equivalent circuit to facilitate the metasurface engineering step.